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I loved learning to read Japanese through the second volume of Heisig's _Learning the Kanji_. Volume 1, which teaches only meanings, is a slog, but volume 2, which teaches the Sino-Japanese readings is a beautiful example of organizing material to minimize entropy and maximize benefit for memorization as soon as possible. Unfortunately he never put together a volume 2 for a Chinese language. I haven't worked on it in a while, but I have a project where I attempt re-create the book for Japanese as well as Mandarin, Korean, and Vietnamese: https://nateglenn.com/uniunihan-db/ (repo: https://github.com/garfieldnate/uniunihan-db).
The "pure groups" are the ones where the presence of a specific radical guarantee you a specific pronunciation (within the list of character/pronunciation pairs you're trying to learn). Of the 4800 characters I used for the volume, only 290 are in the chapter on pure groups. The rest are either in semi-regular groups with varying numbers of exceptions, or in completely irregular groups with no discernible patterns.
The characters were designed continuously over a period of time starting thousands of years ago, and the phonetic parts were sometimes exact and sometimes just clues, similar sounds or rhymes to give the reader a hint. Ancient Chinese pronunciation has changed beyond recognition, so it makes perfect sense that the pronunciations wouldn't be regular anymore.
Mainland China uses a "simplified" character set, which did not affect literacy but in my opinion is a bit more difficult to read; they reduced the number of lines so that more characters look samey and they combined many (Mandarin) homonyms (https://en.wikipedia.org/wiki/Simplified_Chinese_characters#...), removing the meaning portion of characters that would have distinguished them. The simplification did not apply to all characters, so to achieve a high level of literacy you need to know traditional forms, anyway.
It would be interesting to see someone try to actually remodel hanzi from scratch for a specific dialect of Chinese, using 100% regular phonetic components and no variants; multiple pronunciations of a character in the current system would be required to be written differently. An interesting example of this would be certain Korean gukja, where they've combined a Chinese character with a phonetic hangeul (example: https://en.wiktionary.org/wiki/%E3%AB%87). This would be a truly simplified Chinese character set... but all of the culture's history that gets built into spelling over time would be completely lost, which is why I always prefer conservative spelling systems.